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On Trace Sets of Restricted Continued Fraction Semigroups

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Abstract

We record an argument due to Jean Bourgain which gives lower bounds on the size of the trace sets of certain semigroups related to continued fractions on finite alphabets. These bounds are motivated by the “Classical Arithmetic Chaos” Conjecture of McMullen (Dynamics of units and packing constants of ideals, 2012). Specifically, a power is gained in the asymptotic size of the trace set over a “trivial” exponent. The proof involves a new application of the Balog-Szemerédi-Gowers Lemma from additive combinatorics.

Dedicated to the memory of Jean Bourgain

The author is partially supported by an NSF grant DMS-1802119, and the Simons Foundation through MoMath’s Distinguished Visiting Professorship for the Public Dissemination of Mathematics.

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Notes

  1. 1.

    Heath-Brown [19] was later able to do the same for the even thinner polynomial x3 + 2y3, which takes about X2∕3 values up to X.

  2. 2.

    See also [25] for a simpler instance of this “parity breaking.”

  3. 3.

    It turns out that I should have been able to product R-almost primes with R = 7, see [21].

  4. 4.

    Much later, I would exploit a similar feature in [29].

  5. 5.

    For a formal definition of thinness in a general context, see [27, p. 954].

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University, Piscataway, NJ, USA

    Alex Kontorovich

  2. National Museum of Mathematics, New York, NY, USA

    Alex Kontorovich

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Correspondence to Alex Kontorovich .

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  1. Institute of Mathematics, University of Zurich, Zürich, Switzerland

    Artur Avila

  2. Department of Mathematics and Engineering Sciences, Hellenic Military Academy, Athens, Greece

    Michael Th. Rassias

  3. Department of Mathematics, Princeton University, Princeton, NJ, USA

    Yakov Sinai

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Kontorovich, A. (2022). On Trace Sets of Restricted Continued Fraction Semigroups. In: Avila, A., Rassias, M.T., Sinai, Y. (eds) Analysis at Large. Springer, Cham. https://doi.org/10.1007/978-3-031-05331-3_11

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